This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A262244 #38 Aug 10 2025 17:15:00 %S A262244 0,0,1,22,11,1319,25858 %N A262244 Number of concave equilateral n-gons with corner angles of m*Pi/n (0 < m < 2n), where m and n are integers. %C A262244 An n-gon is a polygon with n corners and n sides, each of which is a straight line segment joining two corners. A polygon P is said to be a simple polygon (or a Jordan polygon) if the only points of the plane belonging to multiple edges of P are the corners of P. Such a polygon has a well-defined interior and exterior. Simple polygons are topologically equivalent to a disk, hence zero angles are not allowed; allowable angles are m*Pi/n (where m and n are integers and 0 < m < 2n). A simple n-gon is concave iff at least one of its internal angles is greater than Pi, or equivalently m > n for at least one of the corners. The sum of the m-numbers (called angle factors) for the n-gon has to be n*(n-2). They are partitions of n*(n-2) into n parts with largest part n < k < 2n, and as the edges of a polygon form a closed path, the sum of unit vectors defined by the angle coordinates m/Pi is zero. The reason the m-numbers sum to n*(n-2) is that the sum of the interior angles of any n-gon is Pi*(n-2), and as angles are m*Pi/n, n = Pi. %C A262244 Observation: when n is prime, m is odd and m != n. %H A262244 Stuart E Anderson, <a href="http://www.squaring.net/polygons/cve_5-1_3_3_1_7_.pdf">concave pentagon</a> %H A262244 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o6cve.pdf">concave hexagons</a> %H A262244 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o7cve.pdf">concave heptagons</a> %H A262244 Stuart E Anderson, <a href="http://www.squaring.net/polygons/o8concave.pdf">concave octagons</a> %H A262244 Stuart E Anderson, <a href="/A262244/a262244_1.cpp.txt">jk-latest.cpp</a> produces postscript images of convex, concave and intersecting polygons for n, alternatively there is a switch to produce an unsorted list of interior angle multiples m for each polygon. There is a switch to select if reflected polygons are counted. Polygons produced are all invariant under rotation, new version, Aug 04 2024. %e A262244 For n = 5, the a(5) = 1 solution is (1 3 3 1 7) in m angle factors. %e A262244 For n = 7, the a(7) = 11 solutions in m angle factors are as follows: (1 11 5 3 5 5 5), (1 5 3 9 1 5 11), (1 5 5 1 11 1 11), (1 5 5 5 1 9 9), (1 5 5 5 3 5 11), (1 9 1 9 3 3 9), (1 9 3 5 1 11 5), (1 9 3 5 5 3 9), (3 3 5 5 3 3 13), (3 3 9 3 5 3 9), (3 5 5 5 5 3 9). %Y A262244 Cf. A262181 (convex equilateral polygons). %K A262244 nonn,hard,more %O A262244 3,4 %A A262244 _Stuart E Anderson_, Sep 15 2015 %E A262244 a(9) corrected by _Stuart E Anderson_, Aug 04 2024